Global well-posedness for Keller–Segel system in Besov type spaces

The Cauchy problems for Keller–Segel system are studied using homogeneous Besov spaces. With the homogeneous Besov spaces B ˙ p , ∞ − 2 + n p ( R n ) , which is the scaling critical case for Keller–Segel system, global solutions for small initial data are obtained in the space. In addition, ill-posedness for Keller–Segel system is also studied.